Lineability, differentiable functions and special derivatives

Abstract

The present work either extends or improves several results on lineability of differentiable functions and derivatives enjoying certain special properties. Among many other results, we show that there exist large algebraic structures inside the following sets of special functions: (i.-) The class of differentiable functions with discontinuous derivative on a set of positive measure, (ii.-) the family of differentiable functions with a bounded, non-Riemann integrable derivative, (iii.-) the family of functions from (0, 1) to R that are not derivatives, or (iv.-) the family of mappings that do not satisfy Rolle's Theorem on real infinite dimensional Banach spaces. Several examples and graphics illustrate the obtained results.

Lineability, spaceability, and latticeability of subsets of C([0, 1]) and Sobolev spaces

“…analogously to f the new operator ˆ f is well-defined, linear, injective, and Lipschitz continuous with Lipschitz constant L = 1 (but not an isometry). Using ˆ f we can show the subsequent result (Theorem 2.3 combined with Corollary 2.1 in [18])-thereby D dis denotes the set of all functions h ∈ C([0, 1]) which are differentiable on [0, 1] (at 0 and 1 we consider the one-sided derivates) with a derivative that is discontinuous at every point of a set with positive λ-measure, and D ¬R the family of all functions h ∈ C([0, 1]) which are differentiable on [0, 1] with a derivative that is bounded but not Riemann integrable.…”

“…Considering yet another small modification of f allows for quick alternative proofs for some of the results going back to [18,19]. In fact, setting…”

“…Proof (i) The assertion concerning D dis can be proved as follows: Let f h denote one of the functions constructed in the proof of Theorem 2.3. in [18] and proceed as follows. Defining…”

“…Furthermore, in 2005 [1], he proved that the set of differentiable nowhere monotone functions also contains (except for {0}) infinite dimensional linear spaces. After these seminal works, a lot has been done linking many areas of mathematics, such as Set Theory [8,9,11], Real and Complex Analysis [10,18], Linear and Multilinear Algebra [5], Linear Dynamics [15], or Statistics [12]. Let us recall some terminology we shall need throughout this work (which can be found in [3,4,22,24]).…”

Classical vs. non-Archimedean analysis: an approach via algebraic genericity

“…For instance, we can name algebrability and strong algebrability defined in [5,8], respectively. We refer the interested reader to [1,2,[4][5][6][10][11][12][13][14][14][15][16][17][19][20][21][22]24,25,[27][28][29]45] for a current state of the art on this topic.…”